Let's break down the figures given in the exercise and analyze their rotational symmetry and reflectional symmetry.

13. Identify whether these figures have rotational symmetry, reflectional symmetry, or both. Also, identify the angles of rotational symmetry and lines of reflectional symmetry.

a. Isosceles Triangle

• Rotational Symmetry: No rotational symmetry, because there is no angle (less than 360°) where the triangle maps onto itself other than 360°.
• Reflectional Symmetry: It has one line of reflectional symmetry, which is the vertical line passing through the top vertex and the midpoint of the base.

b. Equilateral Triangle

• Rotational Symmetry: Yes, it has rotational symmetry. It can be rotated by 120° or 240° and still map onto itself. These are the angles of rotational symmetry.
• Reflectional Symmetry: It has 3 lines of reflectional symmetry, one for each vertex to the midpoint of the opposite side.

c. Rhombus

• Rotational Symmetry: Yes, it has rotational symmetry. The rhombus can be rotated by 180° and still map onto itself. The angle of rotational symmetry is 180°.
• Reflectional Symmetry: It has 2 lines of reflectional symmetry: one along the diagonals.

d. Regular Hexagon

• Rotational Symmetry: Yes, it has rotational symmetry. The hexagon can be rotated by 60°, 120°, 180°, 240°, or 300° and map onto itself. These are the angles of rotational symmetry.
• Reflectional Symmetry: It has 6 lines of reflectional symmetry, each one passing through opposite vertices or the midpoints of opposite sides.

To summarize:

• Isosceles Triangle: Reflectional symmetry (1 line), no rotational symmetry.
• Equilateral Triangle: Both reflectional (3 lines) and rotational symmetry (120°, 240°).
• Rhombus: Both reflectional (2 lines) and rotational symmetry (180°).
• Regular Hexagon: Both reflectional (6 lines) and rotational symmetry (60°, 120°, 180°, 240°, 300°).

Would you like to dive deeper into any of these concepts or need further clarification?

Here are five related questions to explore further:

1. What is the difference between rotational symmetry and reflectional symmetry?
2. How would you calculate the number of lines of symmetry for any regular polygon?
3. Can irregular polygons have rotational symmetry? Why or why not?
4. Why does the regular hexagon have more angles of rotational symmetry than the triangle?
5. How does the order of rotational symmetry relate to the number of sides in a regular polygon?

Tip: The order of rotational symmetry is the number of times a figure maps onto itself during a 360° rotation, and the angle is found by dividing 360° by the order of symmetry.